部分双曲系统的拓扑性质与熵的研究

部分双曲系统是对一致双曲系统的一种自然扩展，有着重要的理论意义和极大的实际应用价值。所研究的系统既有一致双曲方向，又有中心方向。中心方向的加入为系统带来了更加复杂的几何结构和动力性态，新的研究工具和技术应运而生，漂亮而艰深的结果不时涌现。部分双曲系统的研究已成为微分动力系统和光滑遍历论引人关注的热点内容。 本项目将侧重研究部分双曲系统的拓扑性质和熵的理论，具体内容包括如下几个方面。 1．研究部分双曲微分同胚在C^0和C^1扰动下不同层次的拟稳定性问题。 2．研究部分双曲微分同胚的拟跟踪性问题。 3. 利用上述拟稳定性和拟跟踪性的结果，研究部分双曲微分同胚的熵的稳定性和连续性等问题。 4. 研究部分双曲微分同胚在随机扰动下的随机拟稳定性问题。

Partial hyperbolicity is a natural extention of the uniform hyperbolicity. So the research for such systems has great significance in theory and application. Since such systems include both hyperbolic direction and center direction, their geometric structure and dynamical behavior are much more complicated. Many new and powerful research tools and techniques appear and many elegant and important results are springing up. The study of partially hyperbolic system has become one of the the main parts of differential dynamical system and smooth ergodic theory. The main task of this project is to study the topological property and entropy of partially hyperbolic systems. It consists of four items. 1. To study the quasi-stability of partially hyperbolic diffeomorphism under C^0 or C^1 perturbation. 2. To study the quasi-shadowing of partially hyperbolic diffeomorphisms. 3. Apply the above results to study the stability and continuity of entropy. 4. To study the quasi-stability of partially hyperbolic diffeomorphism under random perturbation.